In fluid mechanics, or more generally continuum mechanics, incompressible flow is a flow in which the material density does not vary over time. Equivalently, the divergence of an incompressible flow velocity is zero. Under certain conditions, the flow of compressible fluids can be modelled as incompressible flow to a good approximation.
Derivation The fundamental requirement for incompressible flow is that the density, ρ {\displaystyle \rho } , is constant within a small element volume, dV, which moves at the flow velocity u. Mathematically, this constraint implies that the material derivative (discussed below) of the density must vanish to ensure incompressible flow. Before introducing this constraint, we must apply the conservation of mass to generate the necessary relations. The mass is calculated by a volume integral of the density, ρ {\displaystyle \rho } :
m = ∭ V ρ d V . {\displaystyle {m}={\iiint \limits _{V}\!\rho \,\mathrm {d} V}.}
The conservation of mass requires that the time derivative of the mass inside a control volume be equal to the mass flux, J, across its boundaries. Mathematically, we can represent this constraint in terms of a surface integral:
∂ m ∂ t = − ∮ S J ⋅ d S {\displaystyle {\partial m \over \partial t}=-\oint _{S}\mathbf {J} \cdot d\mathbf {S} }
The negative sign in the above expression ensures that outward flow results in a decrease in the mass with respect to time, using the convention that the surface area vector points outward. Now, using the divergence theorem we can derive the relationship between the flux and the partial time derivative of the density:
∭ V ∂ ρ ∂ t d V = − ∭ V ( ∇ ⋅ J ) d V , {\displaystyle {\iiint \limits _{V}{\partial \rho \over \partial t}\,\mathrm {d} V}={-\iiint \limits _{V}\left(\nabla \cdot \mathbf {J} \right)\,\mathrm {d} V},}
therefore:
∂ ρ ∂ t = − ∇ ⋅ J . {\displaystyle {\partial \rho \over \partial t}=-\nabla \cdot \mathbf {J} .}
The partial derivative of the density with respect to time need not vanish to ensure incompressible flow. When we speak of the partial derivative of the density with respect to time, we refer to this rate of change within a control volume of fixed position. By letting the partial time derivative of the density be non-zero, we are not restricting ourselves to incompressible fluids, because the density can change as observed from a fixed position as fluid flows through the control volume. This approach maintains generality, and not requiring that the partial time derivative of the density vanish illustrates that incompressible fluids can still undergo compressible flow. What interests us is the change in density of a control volume that moves along with the flow velocity, u. The flux is related to the flow velocity through the following function:
J = ρ u . {\displaystyle {\mathbf {J} }={\rho \mathbf {u} }.}
So that the conservation of mass implies that:
∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = ∂ ρ ∂ t + ∇ ρ ⋅ u + ρ ( ∇ ⋅ u ) = 0. {\displaystyle {\partial \rho \over \partial t}+{\nabla \cdot \left(\rho \mathbf {u} \right)}={\partial \rho \over \partial t}+{\nabla \rho \cdot \mathbf {u} }+{\rho \left(\nabla \cdot \mathbf {u} \right)}=0.}
The previous relation (where we have used the appropriate product rule) is known as the continuity equation. Now, we need the following relation about the total derivative of the density (where we apply the chain rule):
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